Article

A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions

Adrien Blanchet, and Jérôme Bolte

Abstract

For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Łojasiewicz inequalities. We also discuss the more general case of λ-convex functions and we provide a general convergence theorem for the corresponding gradient dynamics. Specialising our results to the Boltzmann entropy, we recover Otto-Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities. The choice of power-type entropies shows a new equivalence between Gagliardo-Nirenberg inequality and a nonlinear Talagrand inequality. Some nonconvex results and other types of equivalences are discussed.

Keywords

Lojasiewicz inequality; Functional inequalities; Gradient flows; Optimal Transport; Monge-Kantorovich distance;

Replaces

Adrien Blanchet, and Jérôme Bolte, A family of functional inequalities: lojasiewicz inequalities and displacement convex functions, TSE Working Paper, n. 17-787, March 2017.

Reference

Adrien Blanchet, and Jérôme Bolte, A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions, Journal of Functional Analysis, 2018, forthcoming.

Published in

Journal of Functional Analysis, 2018, forthcoming